I would hope to think creativity can be trained. Just like aptitude, one can be born with more or less of it, but I believe exposure and training can greatly enhance or reduce creativity. Research has shown that children became less creative as they go through formal schooling. Something is wrong with this picture.
Creativity is underlying cognitive processes, from conscious learning to automatization.
Our working memory works as a buffer between what has been stored (everything we have experienced) and what is relevant to the task at hand. The brain is working hard, but most of this process goes unnoticed by all of us. Once one can drive for a year or two, one usually uses much less conscious thoughts while driving. Musicians all know how sight reading skills mean the efficient use of their training, or enculturated musical knowledge such as expected harmonic progressions, voice leading and recognition of rhythmic or structural sequences.
Creativity can be deliberate or spontaneous, cognitive or emotional.
If you are interested in the brain, click on the button below to read more about how creativity in the brain has been researched.